Jeremy Lovejoy’s research while affiliated with Université Paris Cité and other places

In his lost notebook, Ramanujan defined the function ϕ ( q ) \phi (q) which is a mock modular form and is related to some of Ramanujan’s mock theta functions. In recent years, a number of congruences for the coefficients of ϕ ( q ) \phi (q) have been proved by Baruah and Begum, Chan, Du and Tang, and Xia. Motivated by their works, we characterize congruences modulo 2 and 4 for the coefficients of ϕ ( q ) \phi (q) based on the congruences on the eighth order mock theta function established by Chen and Garvan. We also prove some congruences modulo 8 for the coefficients of ϕ ( q ) \phi (q) based on an identity due to Newman.

We show how Bailey pairs can be used to give a simple proof of an identity of Chern, Li, Stanton, Xue, and Yee. The same method yields a number of related identities as well as false theta companions.

We prove general fomulas for the deviations of two overpartition ranks from the average. These formulas are in terms of Appell–Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the numerous overpartition rank difference identities in the literature. We give two illustrations.

In this paper we study odd unimodal and odd strongly unimodal sequences. We use q-series methods to find several fundamental generating functions. Employing the Euler--Maclaurin summation formula we obtain the asymptotic main term for both types of sequences. We also find families of congruences modulo 4 for the number of odd strongly unimodal sequences.

We prove general fomulas for the deviations of two overpartition ranks from the average, namely \begin{equation*} \overline{D}(a, M) := \sum_{n \geq 0} \Bigl( \overline{N}(a, M, n) - \frac{\overline{p}(n)}{M} \Bigr) q^n \end{equation*} and \begin{equation*} \overline{D}_{2}(a,M) := \sum_{n \geq 0} \Bigl( \overline{N}_{2}(a, M, n) - \frac{\overline{p}(n)}{M} \Bigr) q^n \end{equation*} where $\overline{N}(a, M, n)$ denotes the number of overpartitions of n with rank congruent to a modulo M, $\overline{N}_{2}(a, M, n)$ is the number of overpartitions of n with $M_2$-rank congruent to a modulo M and $\overline{p}(n)$ is the number of overpartitions of n. These formulas are in terms of Appell-Lerch series and sums of quotients of theta functions and can be used, among other things, to recover any of the numerous overpartition rank difference identities in the literature. We give examples for M=3 and 6.

We construct families of Bailey pairs (αn,βn)$(\alpha _n,\beta _n)$ where the exponent of q in αn$\alpha _n$ is an indefinite quadratic form, but where the usual (-1)j$(-1)^j$ is replaced by a sign function. This leads to identities involving “false” indefinite binary theta series. These closely resemble q-identities for mock theta functions or Maass waveforms, but the sign function prevents them from having the usual modular properties.

Zagier introduced the term "strange identity" to describe an asymptotic relation between a certain q-hypergeometric series and a partial theta function at roots of unity. We show that behind Zagier's strange identity lies a statement about Bailey pairs. Using the iterative machinery of Bailey pairs then leads to many families of multisum strange identities, including Hikami's generalization of Zagier's identity.

As analytic statements, classical q-series identities are equalities between power series for $|q|<1$. This paper concerns a different kind of identity, which we call a quantum q-series identity. By a quantum q-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum q-series identities can all be easily deduced from one simple classical q-series transformation. We then use other results from the theory of q-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.

In a study of congruences for the Fishburn numbers, Andrews and Sellers observed empirically that certain polynomials appearing in the dissections of the partial sums of the Kontsevich–Zagier series are divisible by a certain q-factorial. This was proved by the first two authors. In this paper, we extend this strong divisibility property to two generic families of q-hypergeometric series which, like the Kontsevich–Zagier series, agree asymptotically with partial theta functions.